Application of Derivatives

Find the equation of the tangent and normal to the curve x = a t 2 , y = 2 at at the point ‘t’. - Wired Faculty

Question

Find the equation of the tangent and normal to the curve x = a t² , y = 2 at at the point ‘t’.

Solution

The equations of curve are x = at² , y = 2at
$therefore space space space dx over dt space equals space 2 at comma space space dy over dt equals 2 straight a space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space dy over dx space equals space fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals space fraction numerator 2 straight a over denominator 2 at end fraction space equals 1 over straight t$
$therefore space space space space slope space of space tangent space space equals space 1 over straight t$
$therefore space space space space space the space equation space of space tangent space at space left parenthesis at squared comma space space 2 at right parenthesis space is space space space space space space space space space space space space space space straight y minus 2 at space equals space 1 over straight t left parenthesis straight x minus at squared right parenthesis space space space space space or space space space space ty minus 2 at squared space equals space straight x minus at squared space space space space or space space space space straight x minus ty plus at squared space equals space 0 Also space slope space of space normal space space equals space minus fraction numerator 1 over denominator begin display style 1 over straight t end style end fraction space equals space minus straight t therefore space space space space space equation space of space normal space at space left parenthesis at squared comma space space space 2 at right parenthesis space is space space space space space space space space space space space space space space space space space space straight y minus 2 at space equals space minus straight t left parenthesis straight x minus at squared right parenthesis space or space space space space space space space space space space straight y minus 2 at space equals space minus tx plus at cubed space space space space space space space space space space space or space space tx plus straight y minus 2 at minus at cubed space equals space 0$

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